We prove that given any small but fixed η > 0, a positive proportion of all gaps between consecutive primes are smaller than η times the average gap. We show some unconditional and conditional quantitative results in this vein. In the results the dependence on η is given explicitly, providing a new quantitative way, in addition to that of the first paper in this series, of measuring the effect of the knowledge on the level of distribution of primes.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-1-3,
author = {Daniel Alan Goldston and J\'anos Pintz and Cem Yal\c c\i n Y\i ld\i r\i m},
title = {Primes in tuples IV: Density of small gaps between consecutive primes},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {37-53},
zbl = {1332.11086},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-1-3}
}
Daniel Alan Goldston; János Pintz; Cem Yalçın Yıldırım. Primes in tuples IV: Density of small gaps between consecutive primes. Acta Arithmetica, Tome 161 (2013) pp. 37-53. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa160-1-3/